Projection Transformations

[Hearn & Baker: p. 432-441 (Warning! their perspective projection uses a different eye point!)
Foley & van Dam: p. 229-242 ]

Introduction to Projection Transformations

Mapping:

Projection: Mapping with m less than n
Planar Projection: maps 3D object onto plane

Parallel Projection

parallel lines preserve relative length

Perspective Projection

produces foreshortening
non-linear

A Taxonomy of Projections

A Basic Projection Matrix

Let's use the following example to construct a matrix that performs a perspective transformation:

From similar triangles, we can see that y'/d = y/z
Thus, y' = yd/z
Similarly, x' = xd/z

We can express this in matrix form by altering the value of the homogeneous parameter, h:

[ x ] [1 0 0 0] [x] [ y ] = [0 1 0 0] [y] [ z ] [0 0 1 0] [z] [z/d] [0 0 1/d 0] [1]

non-linear because of divide by h
can incorporate scaling